ملخص المقال

We consider the non-local system with singular non-linearity and singular weights:
\\begin{equation*}
\\left\\lbrace
\\begin{aligned}
&(-\\Delta)^{s}_{p} u + v \\, u^{p - 1} = h(x) u^{-\\alpha},\\, &u > 0 \\quad \\text{in }\\, \\Omega ;\\quad &u = 0, \\quad \\text{in }\\, \\mathbb{R}^{N}\\setminus \\Omega,\\\\[4pt]
&(-\\Delta)^{s}_{p} v = u^{p}, \\, &v > 0 \\quad \\text{in }\\, \\Omega ;\\quad &v = 0, \\quad \\text{in }\\, \\mathbb{R}^{N}\\setminus \\Omega,
\\end{aligned}\\right.
\\end{equation*}
where \\( 0 < s < 1 \\), \\( p > 1 \\), \\( \\alpha > 0 \\), and \\( \\Omega \\subset \\mathbb{R}^{N} \\), with \\( N > sp \\), is an open bounded domain with \\( C^{1,1} \\) boundary \\( \\partial\\Omega \\). The function \\( h : \\Omega \\to \\mathbb{R}^{+} \\) exhibits growth of negative powers of the distance function \\( d(x) := \\text{dist}(x, \\partial\\Omega) \\) near the boundary, that is, \\( h(x) \\sim d^{-\\beta}(x) \\) for some \\( \\beta \\geq 0 \\), when \\( x \\) is close to the boundary \\( \\partial\\Omega \\). For \\( \\beta < sp \\), we discuss the existence of a positive weak solution \\( (u, v) \\in W^{s, p}_{\\text{loc}}(\\Omega) \\times W^{s, p}_{\\text{loc}}(\\Omega) \\) using the classical method of regularization and the fixed point theorem together. Indeed, we found some essential uniform a priori estimates for the approximating sequence before proceeding to the limits. Moreover, we address the uniqueness of finite energy solutions, i.e., \\( (u, v) \\in W^{s, p}_{0}(\\Omega) \\times W^{s, p}_{0}(\\Omega) \\), and demonstrate that this solution pair is a saddle point of a suitable functional when \\( \\alpha < 1 \\). We also provide the boundary behavior of the weak solutions in terms of the distance function. Finally, we establish the non-existence of a weak solution for the case where \\( \\beta \\geq sp \\).

معلومات

Item Type:	Journal
Divisions:
ePrint ID:	5446
Date Deposited:	2025-06-30
Further Information:	Google Scholar
URI:	https://www.univ-soukahras.dz/ar/publication/article/5446

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